$L^p$-integrability of the gradient of solutions to quasilinear systems with discontinuous coefficients

Abstract

The Dirichlet problem for a class of quasilinear elliptic systems of equations with small-BMO coefficients in a Reifenberg-flat domain $\Omega$ is considered. The lower-order terms are supposed to satisfy controlled growth conditions in ${\mathbf u}$ and $D\mathbf {u}$. $L^p$-integrability with $p>2$ of $D{\mathbf u}$ is obtained, where $p$ depends explicitly on the data. An analogous result is obtained also for the Cauchy--Dirichlet problem for quasilinear parabolic systems.